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\begin{document}

\title{2-条件期望}
%\institute{上海立信会计金融学院}
\author{{\ppr LQW}}
\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
%\date{{\ppr 2022年12月30日} }

\maketitle

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%\begin{frame}[fragile=singleslide]{1.1.1. }
\begin{frame}{内容提要 }

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\begin{enumerate}
\item[2.1.]  条件概率
\item[2.2.]  随机变量关于一个事件的条件期望
\item[2.6.]  随机变量关于一个离散型随机变量的条件期望
\item[2.10.]  随机变量关于一个 $\sigma$-域的条件期望
\item[2.13.]  随机变量关于另一个随机变量的条件期望
\item[2.15.]  条件期望的性质
\item[2.E.]  条件期望的一些练习题

\end{enumerate}

\end{frame}

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\begin{frame}{2.1.  条件概率}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{itemize}
\item  定义：设事件 $B$ 的概率不等于零，则在事件 $B$ 发生的条件下，\vspace{0.3cm}
\begin{itemize}
\item  事件 $A$ 发生的{\color{red}条件概率}为 $$\mathbb{P}(A\mid B)=\frac{\mathbb{P}(AB)}{\mathbb{P}(B)}.$$
\item  随机变量 $X$ 的{\color{red}条件分布函数}为 $$F(x\mid B) = \frac{\mathbb{P}(X\le x, B)}{\mathbb{P}(B)}.$$
\end{itemize}


\end{itemize}

\end{frame}

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\begin{frame}{2.2. 随机变量关于一个事件的条件期望 }

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\begin{itemize}
\item  {\color{red}定义：设事件 $B$ 的概率不等于零，则在事件 $B$ 发生的条件下，随机变量 $X$ 的条件期望为 
\begin{eqnarray*}
\mathbb{E}(X\mid B) = \frac{\mathbb{E}(XI_B)}{\mathbb{P}(B)}.
\end{eqnarray*}
}

\item  其中 $I_B$ 是“指示随机变量”，它的定义为 
\begin{eqnarray*}
I_B(\omega) = \left\{ \begin{array}{ll}
1, & \omega\in B, \\
0, & \omega\notin B.
\end{array}\right.
\end{eqnarray*}

\item  例子：当 $X=I_A$ 是另一个事件 $A$ 的指示随机变量时，有
\begin{eqnarray*}
\mathbb{E}(X\mid B) = \frac{\mathbb{E}(I_AI_B)}{\mathbb{P}(B)} = \frac{\mathbb{P}(AB)}{\mathbb{P}(B)} =: \mathbb{P}(A\mid B). 
\end{eqnarray*}
\end{itemize}

\end{frame}

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\begin{frame}{2.3. 条件期望 $\mathbb{E}(X\mid B)$ 的计算公式 }

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\begin{itemize}\itemsep1em

\item  若 $X$ 是离散型随机变量，则 
\begin{eqnarray*}
\mathbb{E}(X\mid B) &=& \sum\limits_{k=1}^{\infty} x_k\mathbb{P}(X=x_k\mid B)  \\
&=&  \sum\limits_{k=1}^{\infty} x_k \frac{\mathbb{P}(X=x_k, B)}{\mathbb{P}(B)}. 
\end{eqnarray*}

\item  设样本空间 $\Omega=\mathbb{R}$. 若 $X$ 是连续型随机变量，概率密度函数 $f(x)$, 则 
$$\mathbb{E}(X\mid B) = \frac{1}{\mathbb{P}(B)}\int_B xf(x)dx. $$

\end{itemize}

\end{frame}

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\begin{frame}{2.4. 例子 - 均匀分布的随机变量的条件期望 }

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\begin{itemize}

\item  设随机变量 $X$ 在区间 $\Omega=(0,1]$ 上均匀分布。
设事件 $B_1=(0,1/3]$, $B_2=(1/3,2/3]$, $B_3=(2/3,1]$. 则期望与条件期望为
\begin{eqnarray*}
\mathbb{E}(X) &=& \int_{\mathbb{R}} xf(x)dx = \int_{0}^{1} x\cdot 1 dx = \frac{1}{2}, \\  
\mathbb{E}(X\mid B_1) &=& \frac{1}{P(B_1)} \int_{B_1} xf(x)dx = \frac{1}{1/3}\int_{0}^{1/3} x\cdot 1 dx = \frac{1}{6}, \\ 
\mathbb{E}(X\mid B_2) &=& \frac{1}{P(B_2)} \int_{B_2} xf(x)dx = \frac{1}{1/3}\int_{1/3}^{2/3} x\cdot 1 dx = \frac{1}{2}, \\ 
\mathbb{E}(X\mid B_3) &=& \frac{1}{P(B_3)} \int_{B_3} xf(x)dx = \frac{1}{1/3}\int_{2/3}^{1} x\cdot 1 dx = \frac{5}{6}.  
\end{eqnarray*}

\end{itemize}

\end{frame}

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\begin{frame}{2.5. 例子的插图 }

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\begin{center}
\includegraphics[height=0.6\textheight, width=0.9\textwidth]{conditional-expectation.png}
\end{center}

\end{frame}

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\begin{frame}{2.6. 随机变量关于一个离散型随机变量的条件期望 }

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\begin{itemize}

\item  设 $Y$ 是一个离散型随机变量。设 $Y$ 取不同的值 $y_1,\cdots, y_n,\cdots$, 设 $$B_n=\{\omega\in \Omega: Y(\omega)=y_n\}.$$
则样本空间 $\Omega$ 是事件 $B_1,B_2,\cdots$ 的互不相交的并集。
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|c|c|} \hline 
$\Omega$ 的子集 & $B_1$ & $\cdots$ & $B_n$ & $\cdots$ \\ \hline 
随机变量 $Y$ 的取值 & $y_1$ & $\cdots$ & $y_n$ & $\cdots$ \\ \hline 
条件期望 $\mathbb{E}(X\mid Y)$ 的取值 & $\mathbb{E}(X\mid B_1)$ & $\cdots$ & $\mathbb{E}(X\mid B_n)$ & $\cdots$ \\ \hline 
\end{tabular}
\end{table}

\item  {\color{red}条件期望 $\mathbb{E}(X\mid Y)$ 定义为一个随机变量
\begin{eqnarray*}
\mathbb{E}(X\mid Y) &:& \Omega \to \mathbb{R} \\
&& \omega \mapsto \mathbb{E}\mathbb(X\mid Y)(\omega) = \mathbb{E}(X\mid B_n), \,\,\text{若}\,\, \omega\in B_n.  
\end{eqnarray*}
}
\end{itemize}

\end{frame}

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\begin{frame}{2.7. $\sigma$-域 }

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\begin{itemize}%\itemsep1em

\item  {\color{red}定义：样本空间 $\Omega$ 上的一个 $\sigma$-域是指一些事件 $B$ 组成的集合 $\mathcal{F}$，符合下述条件：}
\begin{enumerate}
\item  {\color{red}空集与全集一定是 $\mathcal{F}$ 中的元素。}
\item  {\color{red}若事件 $B$ 是 $\mathcal{F}$ 中的元素，则补集 $B^c$ 也一定是 $\mathcal{F}$ 中的元素。}
\item  {\color{red}若可数个事件 $B_1,B_2,\cdots$ 是 $\mathcal{F}$ 中的元素，则它们的并集与交集
$$ \cup_{i=1}^{\infty} B_i, \,\,\,\, \cap_{i=1}^{\infty} B_i $$
也一定是 $\mathcal{F}$ 中的元素。}
\end{enumerate}

\item  例子：$\mathcal{F}_1 = \{\varnothing, \Omega\}$. 
\item  例子：$\mathcal{F}_2 = \{\varnothing, \Omega, B, B^c\}$. 
\item  例子：$\mathcal{F}_3 = \{\varnothing, \Omega, A, B, A^c, B^c, AB, (AB)^c, A\cup B, (A\cup B)^c, \cdots \}$. 

\end{itemize}

\end{frame}

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\begin{frame}{2.8. Borel 域、随机变量的概念}

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\begin{itemize}\itemsep1em
\item  {\color{red}定义：设 $\mathbb{R}$ 是实数全体组成的集合。设 $\mathcal{C}$ 是左开右闭区间全体组成的集合。
包含这个集合的最小的 $\sigma$-域称为是直线上的 Borel 域，记为 $\mathcal{B}$.  }

\item  {\color{red}定义：一个随机变量是指一个可测函数 $X:(\Omega,\mathcal{F}) \to (\mathbb{R}, \mathcal{B})$. }
\item  可测的含义是：对任意 $C\in\mathcal{B}$, 一定有 $X^{-1}(C)\in \mathcal{F}$. 
\item  解释：对任意实数 $a<b$, 事件 $\{a<X\le b\}$ 必须是事件域 $\mathcal{F}$ 中的事件。 
\item  记号：$\{a<X\le b\} = \{\omega: a<X(\omega)\le b\} = X^{-1}((a,b])$
\end{itemize}

\end{frame}

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\begin{frame}{2.9. 由一个随机变量生成的 $\sigma$-域 }

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\begin{itemize}\itemsep1em

\item  {\color{red}定义：由随机变量 $X:(\Omega,\mathcal{F}) \to (\mathbb{R}, \mathcal{B})$ 生成的 $\sigma$-域 $\sigma(X)$ 是指集合
$$\{ \sigma^{-1}(C): C \in \mathcal{B} \}.$$
}

\item  注：由随机变量 $X$ 生成的 $\sigma$-域是包含事件 $\{ X\in (a,b] \}$ 的最小的 $\sigma$-域。

\item  注：由 $X$ 可测可以知道一定有 $\sigma(X)\subseteq \mathcal{F}$. 

\end{itemize}

\end{frame}

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\begin{frame}{2.10. 随机变量关于一个 $\sigma$-域的条件期望 }

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\begin{itemize}%\itemsep1em

\item  设有随机变量 $X:(\Omega,\sigma(X))\to (\mathbb{R},\mathcal{B}). $
\item  设 $\mathcal{F}$ 是样本空间 $\Omega$ 上的一个事件域。
\item  {\color{red}随机变量 $X$ 关于 $\mathcal{F}$ 的条件期望是一个随机变量 $Z$, 满足下述条件。}
\begin{enumerate}\itemsep0.3em
\item   {\color{red}由 $Z$ 生成的 $\sigma$-域 $\sigma(Z)$ 是 $\mathcal{F}$ 的一个子集。}
\item   {\color{red}对任意事件 $B\in\mathcal{F}$, 都成立 $\mathbb{E}(XI_B) = \mathbb{E}(ZI_B). $ }
\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.11. 例子 - 关于平凡事件域的条件期望 }

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\begin{itemize}\itemsep1em

\item  求随机变量 $X:\Omega\to\mathbb{R}$ 关于事件域 $\mathcal{F}=\{\varnothing,\Omega\}$ 的条件期望。
\begin{enumerate}\itemsep0.3em
\item  设这个条件期望为 $Z$.
\item  由 $Z$ 生成的 $\sigma$-域必须是 $\mathcal{F}$ 的子集，所以 $Z$ 必须是常数。
\item  取 $B=\Omega$, 则 $\mathbb{E}(XI_B) = \mathbb{E}(ZI_B)$ 即为 $\mathbb{E}(X) = \mathbb{E}(Z)$. 
\item  因此 $Z$ 只能是 $\mathbb{E}(X)$. 
\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.12. 例子 - 关于一个稍不平凡的事件域的条件期望 }

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\begin{itemize}%\itemsep1em

\item  求随机变量 $X$ 关于事件域 $\mathcal{F}=\{\varnothing,\Omega, B, B^c\}$ 的条件期望。
\begin{enumerate}\itemsep0.3em
\item  设这个条件期望为 $Z$. 
\item  因为 $Z$ 生成的 $\sigma$-域必须是 $\mathcal{F}$ 的子集，所以当 $\omega\in B$ 时 $Z(\omega)$ 是一个常数。%当 $\omega\in B^c$ 时 $Z(\omega)$ 也是一个常数。
\item  考察 $\mathbb{E}(XI_B) = \mathbb{E}(ZI_B)$ 的右边。
\item  当 $\omega\in B$ 时，有 $\mathbb{E}(Z(\omega)I_B) = Z(\omega)\mathbb{P}(B)$. 因此有
\begin{eqnarray*}
Z(\omega) = \frac{\mathbb{E}(XI_B)}{\mathbb{P}(B)} =: \mathbb{E}(X\mid B). 
\end{eqnarray*}

\item  同样地，当 $\omega\in B^c$ 时，有
\begin{eqnarray*}
Z(\omega) = \frac{\mathbb{E}(XI_{B^c})}{\mathbb{P}(B^c)} =: \mathbb{E}(X\mid B^c). 
\end{eqnarray*}

\end{enumerate}


\end{itemize}

\end{frame}

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\begin{frame}{2.13. 一个随机变量关于另一个随机变量的条件期望 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{itemize}%\itemsep1em

\item  设 $X$ 是一个随机变量。设 $Y$ 是另一个随机变量。
\item  {\color{red}定义条件期望 $\mathbb{E}(X\mid Y)$ 是一个随机变量 $Z$, 满足下述条件。}
\begin{enumerate}\itemsep0.3em
\item   {\color{red}由 $Z$ 生成的 $\sigma$-域 $\sigma(Z)$ 是 $\sigma(Y)$ 的一个子集。}
\item   {\color{red}对任意事件 $B\in\sigma(Y)$, 都成立 $\mathbb{E}(XI_B) = \mathbb{E}(ZI_B)$. }
\end{enumerate}

\end{itemize}

\end{frame}


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\begin{frame}{2.14. 条件期望的存在性 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{itemize}%\itemsep1em

\item  设随机变量 $X$ 满足条件 $\mathbb{E}|X|<\infty$, 则对任意事件域 $\mathcal{F}$, 条件期望 $\mathbb{E}(X\mid \mathcal{F})$ 都存在。

\item  证明、例子：

\end{itemize}

\end{frame}

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\begin{frame}{2.15. 条件期望的性质 - 1 }

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\begin{itemize}%\itemsep1em

\item  线性：$\mathbb{E}(c_1X_1+c_2X_2 \mid\mathcal{F} ) = c_1\mathbb{E}(X_1 \mid\mathcal{F}) + c_2\mathbb{E}(X_2 \mid\mathcal{F}). $

\item  证明、例子：


\end{itemize}

\end{frame}


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\begin{frame}{2.16. 条件期望的性质 - 2 }

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\begin{itemize}%\itemsep1em

\item  重期望公式： $\mathbb{E} [ \mathbb{E}(X \mid\mathcal{F}) ] =\mathbb{E}(X)$. 

\item  证明、例子：

\end{itemize}

\end{frame}

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\begin{frame}{2.17. 条件期望的性质 - 3 }

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\begin{itemize}%\itemsep1em

\item  若随机变量 $X$ 与 $\sigma$-域 $\mathcal{F}$ 相互独立，则 $$ \mathbb{E}(X \mid\mathcal{F}) = \mathbb{E}(X). $$ 

\item  若随机变量 $X$ 与随机变量 $Y$ 相互独立，则 $$\mathbb{E}(X \mid Y) = \mathbb{E}(X).$$

\item  证明、例子：

\end{itemize}

\end{frame}

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\begin{frame}{2.18. 条件期望的性质 - 4 }

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\begin{itemize}%\itemsep1em

\item  若随机变量 $X$ 生成的 $\sigma$-域 是事件域 $\mathcal{F}$ 的子集，则 $$\mathbb{E}(X \mid\mathcal{F}) = X. $$ 

\item  证明、例子：


\end{itemize}

\end{frame}

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\begin{frame}{2.19. 条件期望的性质 - 5 }

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\begin{itemize}%\itemsep1em

\item  若随机变量 $X$ 生成的 $\sigma$-域 $\sigma(\mathcal{F})$ 是事件域 $\mathcal{F}$ 的子集，则对任意随机变量 $G$, 都成立（将 $X$ 看作“常数”提出来）
 $$\mathbb{E}(XG \mid\mathcal{F}) = X\mathbb{E}(G \mid\mathcal{F}). $$ 

\item  若随机变量 $X$ 是随机变量 $Y$ 的函数，则对任意随机变量 $G$, 都成立（将 $X$ 看作“常数”提出来）
 $$\mathbb{E}(XG \mid Y) = X\mathbb{E}(G \mid Y ). $$ 

\item  证明、例子：

\end{itemize}

\end{frame}

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\begin{frame}{2.20. 条件期望的性质 - 6 }

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\begin{itemize}%\itemsep1em

\item  若事件域 $\mathcal{F}$ 是另一个事件域 $\mathcal{F}'$ 的子集，则 
\begin{eqnarray*}
\mathbb{E} \left[ \mathbb{E}(X \mid\mathcal{F}') \mid \mathcal{F} \right] &=& \mathbb{E}(X \mid\mathcal{F} ). \\ 
\mathbb{E} \left[ \mathbb{E}(X \mid\mathcal{F}) \mid \mathcal{F}' \right] &=& \mathbb{E}(X \mid\mathcal{F} ). 
\end{eqnarray*}

\item  证明、例子：


\end{itemize}

\end{frame}

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\begin{frame}{2.E.1. 多项选择 }

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\begin{itemize}
\item  %第9题 多项选择 1.4. Conditional Expectation
Find the correct statements about the conditional probability of the event $A$ given the event $B$. 
\begin{enumerate}
\item[a.]  The definition of conditional probability is $P(A|B) = \frac{P(A\cap B)}{P(B)}$. 
\item[b.]  $P(A|B)=P(A)$ if and only if $A$ and $B$ are independent. 
\item[c.]  In the definition of conditional probability, it is crucial that $P(B)$ is positive. 
\item[d.]  The probability measures $P(\cdot)$ and $P(\cdot | B)$ are always two different measures on the sample space $\Omega$. 
\end{enumerate} 

\vspace{0.2cm}

\item  {\color{red}解答：abc. 在事件 $A$ 与 $B$ 相互独立的时候，概率测度 $P(\cdot)$ 与 $P(\cdot | B)$ 是一样的。在事件 $A$ 与 $B$ 不独立的时候，这两个概率测度是不一样的。

}


\end{itemize}

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\begin{frame}{2.E.2. 单项选择 }

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\begin{itemize}
\item  %第10题 单项选择
Roll a die and record the number. Let $B$ be the event that the number is neither one nor two. 
Let $F(x | B)$ be the conditional distribution function given the event $B$. Find $F(5 | B)$. 
\begin{enumerate}
\item[a.]  $5/6$.
\item[b.]  $2/5$.
\item[c.]  $3/4$.
\item[d.]  $2/3$. 
\end{enumerate} 

\vspace{0.2cm}

\item  {\color{red}解答：c. 条件分布函数的定义为 $F(x | B) = \frac{P(X\le x, B)}{P(B)}$. 现在事件 $B=\{3,4,5,6\}$. 因此
$$P(5 | B) = \frac{P(X\le 5, B)}{P(B)} = \frac{P\{3,4,5\}}{P\{3,4,5,6\}} = \frac{3/6}{4/6} = \frac{3}{4}. $$ 

}

\end{itemize}

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\begin{frame}{2.E.3. 单项选择 }

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\begin{itemize}
\item  %第11题 单项选择
Roll a die and let a random variable $X$ record the number. Let $B$ be the event that $X\ge 3$. 
Let $\mathbb{E}(X | B)$ be the conditional expectation given the event $B$. Find $\mathbb{E}(X | B)$. 

\begin{enumerate}
\item[a.]  $7/2$.
\item[b.]  $4$.
\item[c.]  $9/2$.
\item[d.]  $5$.
\end{enumerate} 

\vspace{0.2cm}

\item   {\color{red}解答：c. 随机变量 $X$ 在事件 $B$ 发生的条件下的条件期望的定义是 $$\mathbb{E}(X | B) = \frac{\mathbb{E}(XI_B)}{P(B)},$$ 其中事件 $B$ 的指示函数$I_B$ 当 $\omega\in B$ 时 $I_B(\omega) = 1$, 当 $\omega\notin B$ 时 $I_B(\omega) = 0$. 
 }
 
\end{itemize}

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\begin{itemize}

\item  {\color{red}随机变量 $X$ 与 $XI_B$ 的分布律为
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|} \hline 
$X$ &1&2&3&4&5&6 \\ \hline 
$XI_B$ &0&0&3&4&5&6 \\ \hline 
概率 &1/6&1/6&1/6&1/6&1/6&1/6 \\ \hline
\end{tabular}
\end{table}
}

\item  {\color{red}因此 $XI_B$ 的数学期望与 $(X | B)$ 的条件期望分别为
\begin{eqnarray*}
\mathbb{E}(XI_B) &=& 3\cdot\frac{1}{6} + 4\cdot\frac{1}{6} + 5\cdot\frac{1}{6} + 6\cdot\frac{1}{6} = 3, \\
\mathbb{E}(X | B) &=& \frac{\mathbb{E}(XI_B)}{P(B)} = \frac{3}{4/6} = \frac{9}{2}. 
\end{eqnarray*}
我们发现这个期望值是 3,4,5,6 的平均值。

}

\end{itemize}

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\begin{itemize}
\item  %第12题 单项选择
Consider a random variable $X(\omega)=\omega$ on the space $\Omega = [0,10]$ with uniform distribution. 
Consider an event $B=(4,8]$. Find the conditional expectation $\mathbb{E}(X | B)$. 
\begin{enumerate}
\item[a.]  $2$.
\item[b.]  $4$.
\item[c.]  $6$.
\item[d.]  $8$.
\end{enumerate} 

\vspace{0.2cm}

\end{itemize}

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\begin{itemize}

\item  {\color{red}解答：c. 这里随机变量 $X$ 是连续型的。因为样本空间 $\Omega$ 上的概率分布是均匀分布，所以 $P(B)=\frac{2}{5}$. 现在计算 $\mathbb{E}(XI_B)$. 注意到 $X$ 的密度函数为 $f(x)=\frac{1}{10},\,\, 0\le x\le 10$, 所以
\begin{eqnarray*}
\mathbb{E}(XI_B) = \int_B xf(x)dx = \int_4^8 x\cdot \frac{1}{10}dx = \frac{12}{5}. 
\end{eqnarray*}
因此所求的条件期望为
\begin{eqnarray*}
\mathbb{E}(X I B) =  \frac{\mathbb{E}(XI_B)}{P(B)} = \frac{12/5}{2/5} = 6. 
\end{eqnarray*}

}

\end{itemize}

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\begin{itemize}
\item  %第13题：单项选择
Consider a continuous random variable $X(\omega)=\omega$ on the space $\Omega = [0,10]$ with uniform distribution. 
Consider a discrete random variable $Y$ and the conditional expectation $\mathbb{E}(X|Y)$. 
Find the values of $z_3$ and $p_3$. 

{\footnotesize 
\begin{table}[ht]
\centering
\begin{tabular}{|M{2cm}|M{2cm}|M{2cm}|M{2.5cm}|} \hline
$\omega$ & $0\le \omega< 3$ & $3\le \omega< 6$ & $6\le \omega\le 10$ \\ \hline
$Y(\omega)$ & $100$ & $200$ & $300$ \\ \hline
$\mathbb{E}(X|Y)(\omega)$ & $z_1$ & $z_2$ & $z_3$ \\ \hline
概率 & $p_1$ & $p_2$ & $p_3$ \\ \hline
\end{tabular}
\end{table}
}

\begin{enumerate}
\item[a.]  $z_3=6$,\,\,\, $p_3=0.3$.  
\item[b.]  $z_3=7$,\,\,\, $p_3=0.3$.  
\item[c.]  $z_3=8$,\,\,\, $p_3=0.4$.  
\item[d.]  $z_3=9$,\,\,\, $p_3=0.4$.  
\end{enumerate} 

\vspace{0.2cm}

\end{itemize}

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\begin{itemize}
\item  %第13题：单项选择

{\color{red}解答：c. 离散型随机变量 $Y$ 将样本空间 $\Omega$ 分成互不相交的三个子集 $A_1=[0,3]$, $A_2=(3,6]$ 与 $A_3=(6,10]$. 
随机变量 $X$ 的概率密度函数为 $f(x)=1/10, 0\le x\le 10$. 按定义，当 $\omega\in A_3$ 时，
\begin{eqnarray*}
\mathbb{E}(X|Y)(\omega) = \frac{1}{P(A_3)} \int_{A_3} xf(x)dx = \frac{1}{4/10}\int_6^{10}x\cdot\frac{1}{10}dx = 8. 
\end{eqnarray*}

}

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\begin{itemize}
\item  %第14题：多项选择
Let $X$ and $Y$ be random variables on the space $\Omega$. Let $\mathbb{E}(X|Y)$ be the conditional expectation. Find the correct statements. 
\begin{enumerate}
\item[a.]  $Z$ is a discrete random variable when $Y$ is a discrete random variable. 
\item[b.]  It has the property $\mathbb{E}[\mathbb{E}(X|Y)] = \mathbb{E}(X)$. 
\item[c.]  The conditional expectation $\mathbb{E}(X|Y)$ is a coarser version of $X$, so it is a function of $X$. 
\item[d.]  If $Y$ is a constant, then $\mathbb{E}(X|Y)=\mathbb{E}(X)$.
\end{enumerate} 

\vspace{0.2cm}

\item  {\color{red}解答：abd. 条件期望 $\mathbb{E}(X|Y)$ 确实是随机变量 $X$ 的一个粗糙的版本，但它不是 $X$ 的函数。
它将随机变量 $Y:\Omega\to\mathbb{R}$ 变成另一个随机变量 $\mathbb{E}(X|Y):\Omega\to\mathbb{R}$, 它是随机变量 $Y$ 的函数。
不同的随机变量 $X$ 给出了不同的函数。

}

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\begin{itemize}
\item  %第15题：多项选择
Consider a sample space $\Omega$. Let $\mathcal{F}$ be a $\sigma$-field on $\Omega$. Find the correct statements. 
\begin{enumerate}
\item[a.]  The $\sigma$-field $\mathcal{F}$ is a collection of subsets of $\Omega$. 
\item[b.]  The $\sigma$-field $\mathcal{F}$ is not empty since the empty set $\varnothing$ and the set $\Omega$ are always in the $\sigma$-field. 
\item[c.]  If a subset $A$ is in the $\sigma$-field $\mathcal{F}$, then its complement $A^c$ is also in $\mathcal{F}$. 
\item[d.]  The union or intersection of a countable subsets in $\mathcal{F}$ are always in $\mathcal{F}$.
\end{enumerate} 

\vspace{0.2cm}

\item  {\color{red}解答：abcd. 这些都是 $\sigma$-域的定义中所要求的。

}

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\begin{itemize}
\item  %第16题：多项选择
Let $\mathcal{F}$ be a $\sigma$-field on the sample space $\Omega$. Let $A,B\in \mathcal{F}$. Find the subsets that belong to $\mathcal{F}$.  
\begin{enumerate}
\item[a.]  $A\cap B$.
\item[b.]  $A\cup B$.
\item[c.]  $A-B$.
\item[d.]  $A\times B$. 
\end{enumerate} 

\vspace{0.2cm}

\item  {\color{red}解答：abc. 笛卡尔乘积集合 $A\times B$ 落在 $\Omega\times\Omega$ 中，超出了样本空间 $\Omega$ 的考虑范围。

}

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\begin{itemize}
\item  %第17题：单项选择
Roll a die and let the random variable $X$ be the number observed. Let $\sigma(X)$ be the $\sigma$-field generated by $X$. 
Find the number of elements in $\sigma(X)$. 
\begin{enumerate}
\item[a.]  6.
\item[b.]  16.
\item[c.]  36.
\item[d.]  64.
\end{enumerate} 

\vspace{0.2cm}

\item  {\color{red}解答：d. 随机变量 $X$ 的可能取值为 $1,2,3,4,5,6$. 因此样本空间为 $\Omega=\{1,2,3,4,5,6\}$. 对每个 $1\le k\le 6$, 事件 $\{X=k\}$ 表示出现数字 $k$, 因此这些事件都在随机变量 $X$ 生成的 $\sigma$-域 $\sigma(X)$ 里。因为 $\sigma$-域在补集、可数并集、可数交集这些运算下是封闭的，所以 $\Omega$ 的每个子集都是 $\sigma(X)$ 中的元素。一共有 $2^6=64$ 个元素。

}

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\begin{itemize}
\item  %第18题：多项选择
Find the correct statements about the Borel sets on $\mathbb{R}$. 
\begin{enumerate}
\item[a.]  The intervals $(a,b]$ are Borel sets. 
\item[b.]  The Borel $\sigma$-field $\mathcal{B}(\mathbb{R})$ is the $\sigma$-field generated by intervals of the form $(a,b]$. 
\item[c.]  Each subset of $\mathbb{R}$ is a Borel set. 
\item[d.]  Let $X:\Omega\to\mathbb{R}$ be a random variable, and $A\in\mathcal{B}(\mathbb{R})$ a Borel set. Then $X^{-1}(A)\in\sigma(X)$. 
\end{enumerate} 

\vspace{0.2cm}

\item  {\color{red}解答：abd. 选项 a 和 b 是 Borel 集的定义。并不是实数集的每个子集都是 Borel 集，因此选项 c 不对。选项 d 是随机变量的数学定义。对任意两个实数 $a<b$, 事件 $\{\omega\in \Omega\mid X(\omega)\in (a,b]\}$ 是这个随机变量所带来的一个事件，因此必须在这个随机变量生成的事件域 $\sigma(X)$ 里。

}

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\begin{itemize}
\item  %第19题：多项选择
Let $B=\{B_t, t\ge 0\}$ be the standard Brownian motion. Let $\mathcal{F}_t$ be the $\sigma$-field generated by $(B_s, 0\le s\le t)$. Find the correct statements. 
\begin{enumerate}
\item[a.]  The $\sigma$-field $\mathcal{F}_t$ is the smallest $\sigma$-field containing the essential information about the stochastic process $B$ up to time $t$. 
\item[b.]  For any $0\le s\le t$ and any two real numbers $a<b$, the subset $A_s(a,b)=\{\omega\mid B_s(\omega)\in (a,b]\}$ is an element in $\mathcal{F}_t$. 
\item[c.]  For any $0\le s_1<s_2\le t$ and any Borel set $C\in\mathcal{B}(\mathbb{R}^2)$, the subset $A_{s_1,s_2}(C)=\{\omega\mid (B_{s_1}(\omega), B_{s_2}(\omega))\in C\}$ is an element in $\mathcal{F}_t$. 
\item[d.]  For any $0\le s_1<s_2<s_3$ and any Borel set $C\in\mathcal{B}(\mathbb{R}^3)$, the subset $A_{s_1,s_2,s_3}(C)=\{\omega\mid (B_{s_1}(\omega), B_{s_2}(\omega), B_{s_3}(\omega))\in C\}$ is an element in $\mathcal{F}_t$. 
\end{enumerate} 

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\begin{itemize}

\item  {\color{red}解答：abc. }

\begin{enumerate}
\item  {\color{red}根据时间的先后，由一个随机过程 $X = \{X_t,t\ge 0\}$ 可以生成的 一系列的 $\sigma$-域 $\{\mathcal{F}_t, t\ge 0\}$, 其中 $\mathcal{F}_t$ 是到时刻 $t$ 为止可能发生的样本路径全体。
}

\item  {\color{red}选项 d 中的三个时刻 $s_1 < s_2 < s_3$ 没有说明是否在时刻 $t$ 或之前，因此这个事件 $A_{s_1,s_2,s_3}(C)$ 不一定落在 $\mathcal{F}_t$ 中。
}

\item  {\color{red}任取时间区间 $[0,t]$ 中的有限个时间点 $s_1 < s_2 < \cdots < s_n$, 任取 $n$-维 Borel 子集 $C$, 事件 $$A_{s_1,\cdots, s_n}(C)=\{\omega\mid (B_{s_1}(\omega), \cdots, B_{s_n}(\omega))\in C\}$$ 的集合生成了这个 $\sigma$-域 $\mathcal{F}_t$. 
}
\end{enumerate}


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\begin{itemize}
\item  %第20题：多项选择
Let $X$ be a random variable and $\mathcal{F}$ a $\sigma$-field. Let $Z=\mathbb{E}(X|\mathcal{F})$ be conditional expectation. Find the correct statements. 
\begin{enumerate}
\item[a.]  The random variable $Z$ does not contain more information than that is contained in $\mathcal{F}$. 
\item[b.]  For all $A\in \mathcal{F}$, the random variable $Z$ satisfies the relation $\mathbb{E}(XI_A) = \mathbb{E}(ZI_A)$. 
\item[c.]  The random variable $Z$ is a coarser version of the original random variable $X$. 
\item[d.]  The conditional expectation $\mathbb{E}(X|Y)$ is defined by $\mathbb{E}(X|\sigma(Y))$. 
\end{enumerate} 

\vspace{0.2cm}

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\begin{itemize}

\item  {\color{red}解答：abcd. 选项 a 与 b 是条件期望的定义中的两个条件。选项 c 是说一个随机变量关于某个事件域的条件期望是一个“分辨率较低”的随机变量。

\item  一个例子，随机变量 $X$ 给出全班每个学生的成绩，事件域对应于班级的一些小组。则条件期望仅给出这些小组的平均成绩。而绝对期望则给出了整个班级的平均成绩。

\item  选项 d 是一个随机变量关于另一个随机变量的条件期望的定义。

}

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\begin{frame}{参考文献}

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\begin{thebibliography}{99}

\bibitem{mikosch} {\ppr Thomas Mikosch}. {\ppr Elementary Stochastic Calculus}. 世界图书出版公司，{\ppr 2009} 年 {\ppr 8} 月第 {\ppr 1} 版。
\bibitem{wangjun} 王军、邵吉光、王娟. 随机过程及其在金融领域中的应用. 清华大学出版社，北京交通大学出版社，{\ppr 2018} 年{\ppr 8} 月第 {\ppr 2} 版。
\bibitem{zhangbo} 张波、商豪. 应用随机过程. 中国人民大学出版社，{\ppr 2016} 年 {\ppr 6} 月第 {\ppr 4} 版。
%\bibitem{karlin} {\ppr Mark A. Pinsky, Samuel Karlin}. {\ppr An Introduction to Stochastic Modeling}. 机械工业出版社，{\ppr 2013} 年 {\ppr 2} 月第 {\ppr 1} 版。

\end{thebibliography}

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